“We’ve had no communication with the Diocese,” Southold Town Supervisor Scott Russell said. “We are very concerned about how this will end.”

Amid rumors the diocese is considering demolishing the church Mr. Russell, joined by members of the town’s Historic Preservation Commission, last week wrote a letter to the diocese requesting to be included in any discussion regarding the property’s future.

“It would be a shame for them to demolish it,” said Jim Grathwohl, chairman of the Southold Town Historic Preservation Commission. “It’s an historic structure in Cutchogue.”

On Tuesday, diocesan spokesman Sean Dolan said, “We are not aware of any plans for the building. We are continuing to evaluate the best use of the building for the parishioners in Cutchogue.”

The church building dates back to the 1870s, gaining parish status in 1901 under Rev. John McKenna. The church is listed as a town landmark, and its historic value has been recognized by the Society for the Preservation of Long Island Antiquities and the state Department of Parks and Recreation.

The church, which originally served the Irish community, held its last Mass in January.

The diocese has said an inspection found windows have been leaking for many years and bricks in the foundation had turned soggy. After discovering that the interior plaster was no longer anchored to the wall, the church closed off the building’s eastern half.

The diocese gave a $2 million estimate for the repairs required to bring the post-and-beam structure up to current safety standards. Mr. Grathwohl called that figure “ridiculous.”

“The church is in a way misleading their parishioners,” he added. “They say the construction is very bad and that’s just not true.”

In addition to Sacred Heart, the parish’s holdings include Our Lady of Mercy primary school immediately to the west of the church, a rectory to the east and both a convent and church hall on the south side of Main Road.

Both Mr. Russell and Mr. Grathwohl said they’d like to discuss possible alternative uses for the church.

“We have a keen interest on keeping Scared Heart a part of Cutchogue,” Mr. Russell said. “We want to help in any way we can.”